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Block Scheduling in Two-stage Outpatient Clinics: Appointment Template Design

Pelin Keşrit, Chelliah Sriskandarajah, Jon M. Stauffer

TL;DR

The paper tackles appointment-template design for two-stage outpatient clinics with heterogeneity and service-time uncertainty, using block scheduling to balance workload between a physician assistant in stage 1 and a physician in stage 2. It proves that minimizing waiting time with no idle time is strongly $NP$-Hard and offers implementable heuristics (MA and RB) that achieve no-idle blocks while keeping patient waiting modest, and extends to planning horizons with repeating blocks. Under stochastic service times, it develops a stochastic-programming/SAA framework and shows the heuristics can match stochastic benchmarks when variability is moderate, with RB delivering favorable trade-offs between provider idle/overtime and patient waiting. The study also compares against FCFA, demonstrates robustness to no-shows via overbooking options, and discusses practical extensions to multi-stage settings, making the approach appealing for real-world outpatient clinics. Overall, the work provides actionable, efficient, and scalable appointment templates that improve provider utilization without excessively increasing patient waiting times, even when service times are uncertain.

Abstract

Increasing the efficiency and effectiveness of the healthcare system is a challenge faced worldwide. Many outpatient clinics have implemented two-stage service systems, with both a physician and physician assistant, to enhance capacity and reduce costs. Some patients only visit a physician assistant while some patients visit both providers depending on their patient type. However, minimizing provider idle time and overtime while reducing patient waiting time is challenging in two-stage service systems. Thus, our objective is to find daily appointment templates, based on block scheduling, that minimize a weighted sum of these metrics. A block schedule divides the overall schedule into several time blocks and assigns patients of different types into each block in proportion to their daily demand to balance the workload throughout the day. Since the problem is shown to be strongly $\mathcal{NP}$-Hard, we develop a heuristic algorithm that provides a no-idle time appointment template that is easily implementable. We expand our study to include stochastic service times and show that our algorithm yields an efficient block schedule under practically relevant conditions. The algorithm is able to provide a solution similar in cost to the stochastic model when patient wait time costs are low by maintaining lower physician idle times with at most a 16 minute/patient increase in patient wait times. Comparing our heuristic to a First Come, First Appointment scheduling rule, we show that our heuristic is able to better minimize provider idle time, which mimics many real-life settings where clinics prioritize the efficiency of the healthcare providers.

Block Scheduling in Two-stage Outpatient Clinics: Appointment Template Design

TL;DR

The paper tackles appointment-template design for two-stage outpatient clinics with heterogeneity and service-time uncertainty, using block scheduling to balance workload between a physician assistant in stage 1 and a physician in stage 2. It proves that minimizing waiting time with no idle time is strongly -Hard and offers implementable heuristics (MA and RB) that achieve no-idle blocks while keeping patient waiting modest, and extends to planning horizons with repeating blocks. Under stochastic service times, it develops a stochastic-programming/SAA framework and shows the heuristics can match stochastic benchmarks when variability is moderate, with RB delivering favorable trade-offs between provider idle/overtime and patient waiting. The study also compares against FCFA, demonstrates robustness to no-shows via overbooking options, and discusses practical extensions to multi-stage settings, making the approach appealing for real-world outpatient clinics. Overall, the work provides actionable, efficient, and scalable appointment templates that improve provider utilization without excessively increasing patient waiting times, even when service times are uncertain.

Abstract

Increasing the efficiency and effectiveness of the healthcare system is a challenge faced worldwide. Many outpatient clinics have implemented two-stage service systems, with both a physician and physician assistant, to enhance capacity and reduce costs. Some patients only visit a physician assistant while some patients visit both providers depending on their patient type. However, minimizing provider idle time and overtime while reducing patient waiting time is challenging in two-stage service systems. Thus, our objective is to find daily appointment templates, based on block scheduling, that minimize a weighted sum of these metrics. A block schedule divides the overall schedule into several time blocks and assigns patients of different types into each block in proportion to their daily demand to balance the workload throughout the day. Since the problem is shown to be strongly -Hard, we develop a heuristic algorithm that provides a no-idle time appointment template that is easily implementable. We expand our study to include stochastic service times and show that our algorithm yields an efficient block schedule under practically relevant conditions. The algorithm is able to provide a solution similar in cost to the stochastic model when patient wait time costs are low by maintaining lower physician idle times with at most a 16 minute/patient increase in patient wait times. Comparing our heuristic to a First Come, First Appointment scheduling rule, we show that our heuristic is able to better minimize provider idle time, which mimics many real-life settings where clinics prioritize the efficiency of the healthcare providers.

Paper Structure

This paper contains 29 sections, 7 theorems, 44 equations, 16 figures, 16 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Research Questions and Contributions
  4. Literature Review
  5. Single-Stage Appointment Scheduling
  6. Multi-stage Appointment Scheduling
  7. Model and Analysis for Two-stage Scheduling of a Block
  8. Two-stage Block Scheduling
  9. Two-stage Scheduling Model for a Block
  10. Single Block Schedule Minimizing System Idle Time
  11. Model and Analysis for Two-stage Scheduling for the Planning Horizon
  12. Scheduling Multiple Blocks for a Planning Horizon
  13. Service Time Uncertainty
  14. Two-stage Scheduling Model for a Block with Stochastic Service Times
  15. Scheduling with Stochastic Service Times
  16. ...and 14 more sections

Key Result

Theorem 1

Minimizing patient waiting time in a block schedule $\pi$ having zero system idle time (Problem $P_1$) is strongly $\mathcal{NP}$-Hard.

Figures (16)

  • Figure 1: Patient Flow Chart
  • Figure 2: A No-Idle Block Schedule $\pi_1$ for Example 1. $\sigma=\{T_3, T_4, T_4, T_4, T_2,T_2,T_1,T_1,T_1\}$ is the patient sequence in $\pi_1$. The total patient waiting time in $\pi_1$ is 90.
  • Figure 3: The Block Schedule $\pi_1$ with $H_1$, $B_1$, $T_1$ for Example 1
  • Figure 4: Example 1: The Schedule $\Gamma= \{\pi_1\pi_2\}$ is a concatenation of two blocks. $\sigma$ is patient sequence in $\pi_1$ and $\pi_2$.
  • Figure 5: An illustration of Steps 1-3 of Algorithm \ref{['alg-B']} for Example 1
  • ...and 11 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6